Question

If α, β, are zeroes of polynomial p(x) = 5x2 + 5x + 1 then find the value of alpha cube + beta cube

Answer 1

❏ Question

If α, β, are zeroes of polynomial p(x) = 5x² + 5x + 1 then find the value of α³ + β³

❏ Solution

Given:-

• polynomial ,p(x) = 5x²+5x+1 = 0
• α and β are zeroes.

Find:-

• Value of α³ + β³

❏ Explanation

Given polynomial,

➥ p(x) = 5x² + 5x + 1 = 0

★Sum of zeroes = -(coefficient of x)/(coefficient if x²)

➥ α + β = -(5)/5

➥ α + β = -1 .............(1)

★ Product of zeroes = -(constant part)/(coefficient of x²)

➥ α . β = 1/5 ...........(2)

Squaring both side of equ(1),

Squaring both sides ,

➥ ( α + β )² = 1²

➥ α² + β² + 2α . β = 1

keep value by equ(2),

➥ α² + β² = 1 - 2×1/5

➥ α² + β² = 1 - 2/ 5

➥ α² + β² = (5-2)/5

➥ α² + β² = 3/5 .............(3)

we know,

★( α³ + β³) = ( α + β)(α² + β²- α . β)

So, keep value by equ(1),(2) and (3)

➥ ( α³ + β³) = (-1). ( 3/5 - 1/5)

➥ ( α³ + β³) = -(3-1)/5

➥ ( α³ + β³) = ( -2/5 ). Ans.

Answer 2

Step-by-step explanation:

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The given quadratic polynomial is

5x^2+5x+1=0

α+β=-b/a

=-5/5

=-1

(Taking cube of both sides)

(α+β)^3=(-1)^3

= -1

αβ=c/a

=1/5

α^3+β^3=(α+β)^3-3α^2β-3αβ^2

= -1-3α^2β-3αβ^2

=-1-3αβ(α+β)

= -1 -3(1/5)(-1)

=-1+3/5

= (-5+3)/5

α^3+β^3= -2/5

The value of α^3+β^3 is -2/5

Hope it helps you:)